After proper rescaling and under some technical assumptions, the smallest eigenvalue of a sample covariance matrix with aspect ratio bounded away from 1 converges to the Tracy-Widom distribution. This complements the results on the largest eigenvalue, due to Soshnikov and Péché.

We compute the mixing rate of a non-backtracking random walk on a regular expander. Using some properties of Chebyshev polynomials of the second kind, we show that this rate may be up to twice as fast as the mixing rate of the simple random walk. The closer the expander is to a Ramanujan graph, the higher the ratio between the above two mixing rates is.As an application, we show that if G is a high-girth regular expander on n vertices, then a typical non-backtracking random walk of length n on G does not visit a vertex more than (1 + o (1)) log n log log n times, and this result is tight. In this sense, the multi-set of visited vertices is analogous to the result of throwing n balls to n bins uniformly, in contrast to the simple random walk on G, which almost surely visits some vertex Ω(log n) times.

We study the asymptotic distribution of the eigenvalues of random Hermitian periodic band matrices, focusing on the spectral edges. The eigenvalues close to the edges converge in distribution to the Airy point process if (and only if) the band is sufficiently wide (W N 5=6 ). Otherwise, a different limiting distribution appears.

We demonstrate a simple analytic argument that may be used to bound the Lévy concentration function of a sum of independent random variables. The main application is a version of a recent inequality due to Rudelson and Vershynin, and its multidimensional generalisation.Des bornes pour la fonction de concentration en matière d'approximation Diophantienne. Nous montrons un simple raisonnement analytique qui peutêtre utile pour borner la fonction de concentration d'une somme des variables aléatoires indépendantes. L'application principale est une version de l'inégalité récente de Rudelson et Vershynin, et sa généralisation au cadre multidimensionel.

We study localisation effects of strong disorder on the spectral and dynamical properties of (matrix and scalar) Schrödinger operators with nonmonotone random potentials, on the d-dimensional lattice. Our results include dynamical localisation, i.e. exponentially decaying bounds on the transition amplitude in the mean. They are derived through the study of fractional moments of the resolvent, which are finite due to resonance-diffusing effects of the disorder. One of the byproducts of the analysis is a nearly optimal Wegner estimate. A particular example of the class of systems covered by our results is the discrete alloy-type Anderson model.where the entries v(n) of the potential are random and independent.The basic phenomenon, named Anderson localisation after the physicist P. W. Anderson, is that disorder can cause localisation of electron states, which manifests itself in time evolution (non-spreading of wave packets), (vanishing of) conductivity in response to electric field, Hall currents in the presence of both magnetic and electric field, and statistics of the spacing between nearby energy levels. The first property implies spectral localisation, i.e. the spectral measure of H A is almost surely pure point, and almost sure exponential decay of eigenfunctions.These properties are known to hold for H A in each of the following cases: 1) high disorder (the coupling constant g is large), 2) extreme energies, 3) weak disorder away from the spectrum of the unperturbed operator, and 4) one dimension, d = 1.Historically, the first proof of spectral localisation was given by Goldsheid, Molchanov and Pastur [12], for a one-dimensional continuous random Schrödinger operator.In higher dimension, the absence of diffusion was first established in 1983 by Fröhlich and Spencer [9] using multi-scale analysis. Their approach has led to a multitude of results on localisation for a wide range of problems. The reader is referred to the monograph of Stollmann [17] or the recent lecture notes of W. Kirsch [14] for a review of the history of the subject and a gentle introduction to the multiscale analysis -which is not used here.One of the ingredients of multi-scale analysis is the regularity of the integrated density of states, the (distribution function of the) average of the spectral measure over the randomness.Ten years later Aizenman and Molchanov [2] introduced an alternative method for the proof of localisation, known as the fractional moment method, which has also found numerous applications. In particular, in [1], Aizenman introduced the notion of eigenfunction correlator, which, combined with the fractional moment method, allowed him to give the first proof of dynamical localisation. We refer to the lecture notes of Stolz [19] and Aizenman and Warzel [3] for a survey of subsequent developments.In the fractional moment method, an a priori estimate on the diagonal elements of the resolvent (H A − λ) −1 plays a key rôle in the underlying analysis.